Proof: By Induction
(related to Corollary: Circular References Of Self-Contained Sets Are Forbidden)
We conduct a proof by induction.
Base Case $n=0$
- We have to show that $X_0\not\in X_0.$
- By the axiom of foundation, the singleton $\{X_0\}$ contains an element that is disjoint from itself.
- This can only be the set $X_0$.
- Since $X_0\cap \{X_0\}=\emptyset$, we have $X_0\not\in X_0.$
Induction step $n\rightarrow n+1$
- By the base case there is no such chain which fulfills $X_0\in X_1\in\ldots\in X_{n-1}\in X_n\in X_0.$
- If a chain $X_0\in X_1\in\ldots\in X_n\in X_{n+1}\in X_0$ existed, then there would be a set $X_{n+1}$ with $X_{n+1}\in X_0$ and $X_n\in X_{n+1}$.
- Therefore, we cound replace $X_n$ by $X_{n+1}$ and get the chain $X_0\in X_1\in\ldots\in X_{n-1}\in X_{n+1}\in X_0.$
- But this is a contradiction to the base case.
∎
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition
